Free functor into a Kleisli category ($F_T$)

The free functor $F_T:\mathcal{C}\to {\mathcal{C}}_T$ is the identity on objects and sends $f:A\to B$ to ${\eta }_{B}f:A\to TB$. A functor $G:\mathcal{C}\to \mathcal{D}$ carries an algebra for the precomposition monad $G\mapsto GT$ exactly when it factors through $F_T$.